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+
+
+.. _sphx_glr_auto_examples_plot_OT_L1_vs_L2.py:
+
+
+==========================================
+2D Optimal transport for different metrics
+==========================================
+
+2D OT on empirical distributio with different gound metric.
+
+Stole the figure idea from Fig. 1 and 2 in
+https://arxiv.org/pdf/1706.07650.pdf
+
+
+
+
+
+.. code-block:: python
+
+
+ # Author: Remi Flamary <remi.flamary@unice.fr>
+ #
+ # License: MIT License
+
+ import numpy as np
+ import matplotlib.pylab as pl
+ import ot
+ import ot.plot
+
+
+
+
+
+
+
+Dataset 1 : uniform sampling
+----------------------------
+
+
+
+.. code-block:: python
+
+
+ n = 20 # nb samples
+ xs = np.zeros((n, 2))
+ xs[:, 0] = np.arange(n) + 1
+ xs[:, 1] = (np.arange(n) + 1) * -0.001 # to make it strictly convex...
+
+ xt = np.zeros((n, 2))
+ xt[:, 1] = np.arange(n) + 1
+
+ a, b = ot.unif(n), ot.unif(n) # uniform distribution on samples
+
+ # loss matrix
+ M1 = ot.dist(xs, xt, metric='euclidean')
+ M1 /= M1.max()
+
+ # loss matrix
+ M2 = ot.dist(xs, xt, metric='sqeuclidean')
+ M2 /= M2.max()
+
+ # loss matrix
+ Mp = np.sqrt(ot.dist(xs, xt, metric='euclidean'))
+ Mp /= Mp.max()
+
+ # Data
+ pl.figure(1, figsize=(7, 3))
+ pl.clf()
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ pl.title('Source and target distributions')
+
+
+ # Cost matrices
+ pl.figure(2, figsize=(7, 3))
+
+ pl.subplot(1, 3, 1)
+ pl.imshow(M1, interpolation='nearest')
+ pl.title('Euclidean cost')
+
+ pl.subplot(1, 3, 2)
+ pl.imshow(M2, interpolation='nearest')
+ pl.title('Squared Euclidean cost')
+
+ pl.subplot(1, 3, 3)
+ pl.imshow(Mp, interpolation='nearest')
+ pl.title('Sqrt Euclidean cost')
+ pl.tight_layout()
+
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+ *
+
+ .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_001.png
+ :scale: 47
+
+ *
+
+ .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_002.png
+ :scale: 47
+
+
+
+
+Dataset 1 : Plot OT Matrices
+----------------------------
+
+
+
+.. code-block:: python
+
+
+
+ #%% EMD
+ G1 = ot.emd(a, b, M1)
+ G2 = ot.emd(a, b, M2)
+ Gp = ot.emd(a, b, Mp)
+
+ # OT matrices
+ pl.figure(3, figsize=(7, 3))
+
+ pl.subplot(1, 3, 1)
+ ot.plot.plot2D_samples_mat(xs, xt, G1, c=[.5, .5, 1])
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ # pl.legend(loc=0)
+ pl.title('OT Euclidean')
+
+ pl.subplot(1, 3, 2)
+ ot.plot.plot2D_samples_mat(xs, xt, G2, c=[.5, .5, 1])
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ # pl.legend(loc=0)
+ pl.title('OT squared Euclidean')
+
+ pl.subplot(1, 3, 3)
+ ot.plot.plot2D_samples_mat(xs, xt, Gp, c=[.5, .5, 1])
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ # pl.legend(loc=0)
+ pl.title('OT sqrt Euclidean')
+ pl.tight_layout()
+
+ pl.show()
+
+
+
+
+
+.. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_005.png
+ :align: center
+
+
+
+
+Dataset 2 : Partial circle
+--------------------------
+
+
+
+.. code-block:: python
+
+
+ n = 50 # nb samples
+ xtot = np.zeros((n + 1, 2))
+ xtot[:, 0] = np.cos(
+ (np.arange(n + 1) + 1.0) * 0.9 / (n + 2) * 2 * np.pi)
+ xtot[:, 1] = np.sin(
+ (np.arange(n + 1) + 1.0) * 0.9 / (n + 2) * 2 * np.pi)
+
+ xs = xtot[:n, :]
+ xt = xtot[1:, :]
+
+ a, b = ot.unif(n), ot.unif(n) # uniform distribution on samples
+
+ # loss matrix
+ M1 = ot.dist(xs, xt, metric='euclidean')
+ M1 /= M1.max()
+
+ # loss matrix
+ M2 = ot.dist(xs, xt, metric='sqeuclidean')
+ M2 /= M2.max()
+
+ # loss matrix
+ Mp = np.sqrt(ot.dist(xs, xt, metric='euclidean'))
+ Mp /= Mp.max()
+
+
+ # Data
+ pl.figure(4, figsize=(7, 3))
+ pl.clf()
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ pl.title('Source and traget distributions')
+
+
+ # Cost matrices
+ pl.figure(5, figsize=(7, 3))
+
+ pl.subplot(1, 3, 1)
+ pl.imshow(M1, interpolation='nearest')
+ pl.title('Euclidean cost')
+
+ pl.subplot(1, 3, 2)
+ pl.imshow(M2, interpolation='nearest')
+ pl.title('Squared Euclidean cost')
+
+ pl.subplot(1, 3, 3)
+ pl.imshow(Mp, interpolation='nearest')
+ pl.title('Sqrt Euclidean cost')
+ pl.tight_layout()
+
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+ *
+
+ .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_007.png
+ :scale: 47
+
+ *
+
+ .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_008.png
+ :scale: 47
+
+
+
+
+Dataset 2 : Plot OT Matrices
+-----------------------------
+
+
+
+.. code-block:: python
+
+
+
+ #%% EMD
+ G1 = ot.emd(a, b, M1)
+ G2 = ot.emd(a, b, M2)
+ Gp = ot.emd(a, b, Mp)
+
+ # OT matrices
+ pl.figure(6, figsize=(7, 3))
+
+ pl.subplot(1, 3, 1)
+ ot.plot.plot2D_samples_mat(xs, xt, G1, c=[.5, .5, 1])
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ # pl.legend(loc=0)
+ pl.title('OT Euclidean')
+
+ pl.subplot(1, 3, 2)
+ ot.plot.plot2D_samples_mat(xs, xt, G2, c=[.5, .5, 1])
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ # pl.legend(loc=0)
+ pl.title('OT squared Euclidean')
+
+ pl.subplot(1, 3, 3)
+ ot.plot.plot2D_samples_mat(xs, xt, Gp, c=[.5, .5, 1])
+ pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+ pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+ pl.axis('equal')
+ # pl.legend(loc=0)
+ pl.title('OT sqrt Euclidean')
+ pl.tight_layout()
+
+ pl.show()
+
+
+
+.. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_011.png
+ :align: center
+
+
+
+
+**Total running time of the script:** ( 0 minutes 0.958 seconds)
+
+
+
+.. only :: html
+
+ .. container:: sphx-glr-footer
+
+
+ .. container:: sphx-glr-download
+
+ :download:`Download Python source code: plot_OT_L1_vs_L2.py <plot_OT_L1_vs_L2.py>`
+
+
+
+ .. container:: sphx-glr-download
+
+ :download:`Download Jupyter notebook: plot_OT_L1_vs_L2.ipynb <plot_OT_L1_vs_L2.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
generated by cgit v1.2.3 (git 2.39.1) at 2024-06-17 15:52:49 +0000